> But, every countable set does have a bijection with N or has a> bijection with some member of w (the set of natural numbers).

That would be true if countability and aleph_0 were not self-contradicting concepts.

But there is a set that is less than uncountable but has no bijectionwith N or a definasble subset of N. This set is the set of all finitedefinitions (in binary representation).

010001...

where every line may be enumerated by an element of the countable setomega^omega^omega (and, if required, finitely many more exponents foralphabets, languages, dictionaries, thesauruses, and furtherproperties)

An obvious enumeration of the lines is 1, 2, 3, ... where every linencan have many sub-enumerations

nn.1.an.11.an.111.a...

Every section of 1's and a's contains enough symbols to enumerateevery single line as often as required to cover all its meanings inall possible languages. If necessary we can add 2's and b's and so onand remain in the countable domain.

This set contains all possible finite words in all possible languagesover all possible alphabets over all possible whatever may be requiredin addition.This set is countable. Hence the set of all definable, computable, and"whatever may be invented by clever set theorists" real numbers iscountable as a subset.

Cantor shows that a countable set is uncountable or that a constructedreal is unconstructable.